Investigation of Tip Clearance Phenomena in an Axial Compressor Cascade Using Euler and Navier-Stokes Procedures

Three-dimensional Euler and Full Navier-Stokes computational procedures have been utilized to simulate the flow field in an axial compressor cascade with tip clearance. An embedded H-grid topology was utilized to resolve the flow physics in the tip gap region. The numerical procedure employed is a finite difference Runge-Kutta scheme. Available measurements of blade static pressure distributions along the blade span, dynamic pressure and flow angle in the cascade outlet region, and spanwise distributions of blade normal force coefficient and circumferentially averaged flow angle are used for comparison. Several parameters which were varied in the experimental investigations were also varied in the computational studies. Specifically, measurements were taken and computations were performed on the configuration with and without: tip clearance, the presence of an endwall, inlet endwall total pressure profiles and simulated relative casing rotation. Additionally, both Euler and Navier-Stokes computations were performed to investigate the relative performance of these approaches in reconciling the physical phenomena considered. Results indicate that the Navier-Stokes procedure, which utilizes a low Reynolds number k-e model, captures a variety of important physical phenomena associated with tip clearance flows with good accuracy. These include tip vortex strength and trajectory, blade loading near the tip, the interaction of the tip clearance flow with passage secondary flow and the effects of relative endwall motion. The Euler computation provides good but somewhat diminished accuracy in resolution of some of these clearance phenomena. It is concluded that the level of modelling embodied in the present approach is sufficient to extract much of the tip region flow field information useful to designers of turbomachinery.Copyright © 1992 by ASME

njss, nips pitchwise indices at suction, pressure surfaces p pressure q total velocity vector 42D total velocity vector at midspan and midpassge s distance along chord t pitch (= 0.152 m) u, v, w cartesian velocity components x, y, z cartesian coordinates (Figure 1

INTRODUCTION
The execution rate performance of efficient CFD codes on modern vector processors has reached 0(10 2) MFLOPS.Researchers can now obtain three-dimensional full Navier-Stokes solutions using 0(105 ) grid points in 0(100) CPU hours.Such capability has offered, among others things, the ability to perform large scale viscous simulation studies to develop an improved understanding of particular physical processes in turbomachines.
The physics of tip clearance flows have received a great deal of attention from both experimental and computational investigators in the gas turbine field.This is primarily due to the importance of complex endwall and clearance phenomena in determining stage performance.The trend towards more highly loaded and lower aspect ratio blades intensifies the importance of these physics to the turbomachinery engineer.
Several of the dominant physical processes involved in tip clearance flows, including blade unloading and vorticity transport, are primarily rotational inviscid in nature.For this reason, several very useful tip clearance models have been devised based on various approximations to the Euler equations, with and without viscous corrections [Rains(1954), Lakshminarayana (1970), Moore and Tilton (1987), Chen et al. (1990), Yaras et al. (1991)].There are also several considerations which warrant the use of viscous analyses to provide more detailed simulation of tip clearance flows.The characteristics of the boundary layers along the blades near the tip, and on the tip and casing surfaces themselves, influence significantly flows with tip clearance.Typical of these phenomena are the often separated flow near the suction surface-casing endwall interface, the influence of relative casing wall rotation and blockage induced by the boundary layer which develops on the blade tip.Also, the generation and decay of vorticity must be modelled (or rely on artificial diffusion) in subsonic inviscid flow computations.Tip clearance effects have been incorporated in viscous turbomachinery flow field computations by numerous researchers in recent years.Often a "thin blade" approximation is used to model the tip geometry.This allows for the convenient specification of a periodic boundary in the tip gap region, while retaining a standard H-grid topology.This approach has been utilized by many investigators including Storer and Cumpsty (1990), Bansod and Rhie (1990), Hah and Reid (1991), Kunz and Lakshminarayana (1990) to name a few recent examples.The grid near the casing becomes highly skewed due to the "pinching" of the blade tip, when this simple approach is used.However, for thin blade profiles this approach has been shown to provide a good engineering approximation when modelling the flow in the tip gap region.
Several alternative grid topologies can be incorporated to directly resolve the tip gap region.Rai (1989), Beach (1990) and Choi and Knight (1991) have used patched 0-H-grid structures, with stacked 0-grids in the gap region.Watanbe et al. (1991) recently implemented an overlapping C-H topology with stacked H-grids in the tip region.An "embedded" H-grid topology has found recent popularity in gas turbine applications with tip clearance [Moore and Moore (1989), Cleak and Gregory-Smith (1991), Liu et al. (1991), Briley et al. (1991), Basson et al. (1991)].All of the mentioned approaches have in common that they provide significantly smoother meshes, more precise geometrical representation and improved flow field resolution in the tip clearance region.
In this paper, an Euler/ Navier-Stokes flow solver has been used to study a variety of tip clearance phenomena.The code, described in detail in Kunz and Lakshminarayana (1991), was recently modified to accommodate the embedded H-grid topology.In earlier work by the present authors [Basson et al. (1991)], the approach was used to simulate the flow in a linear compressor cascade with clearance.Though the emphasis of that work was on grid generation, it was found that the method provided good predictions of near tip blade pressure and wake flow angle distributions when compared with experimental data.Also, it was observed that for the relatively thick compressor cascade blade considered, the embedded H-grid solution provided better agreement with these experimental measurements than a solution which incorporated a "pinched" standard H-grid.
A series of numerical investigations have been performed since that preliminary work, and the results of these are presented and interpreted in this paper.The goals of the present work are: 1) Evaluate the relative performance of Navier-Stokes and Euler analyses in predicting a variety of tip clearance physics.Also, provide some additional results comparing and contrasting the embedded and standard H-grid topologies in viscous flow computations.
2) Perform a series of parametric investigations using the viscous flow code and compare these results with available experimental data.
3) Perform an additional parametric simulation study incorporating the effect of relative endwall motion.4) Draw conclusions, based on the above, on the physical nature of flows with tip clearance and evaluate the capability and limitations of the viscous analysis in reconciling these phenomena.

EXPERIMENTAL AND NUMERICAL PROCEDURES Experimental Procedures
The configuration chosen for computational study in this paper is the so called "Liverpool Cascade."This configuration was the subject of a variety of experimental tip clearance flow investigations undertaken by the second author in earlier work [see Lakshminarayana and Horlock (1965), Lakshminarayana (1970)].Extensive details of the facility are provided in Lakshminarayana and Horlock (1965).A brief overview is provided here.A low speed cascade tunnel in the turbomachinery laboratory at Liverpool University was used for the experimental investigations.The cascade had nine "split" blades of 10C430050 profile, of chord length 0.152m, set at a 36 0 stagger.The blades were cantilevered from a solid frame as shown in Figure 1.One side of the tunnel was movable allowing the distance between the two halves of the split blades, 2i (Figure 1), to be adjusted.The aspect ratio of the unsplit blades was 4.8 and the space-chord ratio was 1.00.For the tests detailed below, the air inlet and outlet angles were 51 0 and 31 0 respectively, and the Reynolds number based on chord length and flow velocity at midspan was 2.0 x 10 5 .Figure 1 provides a schematic of the experimental configuration.Three series of experiments were performed, designated A, B and C in Figure 1.In Experiments A, the blades were separated a distance 2ti and the upstream velocity profile was uniform (Figure la).This served to isolate tip clearance physics without the influence of an endwall or the interaction between passage secondary motion and the tip gap flow.The cascade flow field in Experiment A consists of a row of anti-symmetric leakage vortex pairs.The plane of symmetry of these vortex pairs lies halfway between the tips of the split blades.In Experiments B, a plate was inserted well upstream of the cascade to provide a non-uniform inlet flow (Figure lb).The motive of these studies was to investigate the interaction between leakage flow and the weak secondary flow due to blade to blade turning of an inlet normal vorticity profile.In Experiments C, a thin wall was inserted in the center of the gap (Figure lc) to simulate the presence of an endwall.The endwall gives rise to significantly stronger passage secondary motion, since the inlet vorticity gradient is larger and is sustained by viscous shear at the endwall.Additionally, a boundary layer develops along the endwall in the tip gap which gives rise to additional blockage of the clearance flow.By also running Experiments B and C with zero clearance, Lakshminarayana and Horlock (1965) were able to observe and compare the isolated secondary flows in these cases, without the presence of the tip clearance flow.

Numerical Procedures
A three-dimensional Navier-Stokes flow solver was used for the computational studies presented herein.A complete description of the governing equations and numerical procedures implemented is available in Kunz and Lakshminarayana (1991).A brief overview is provided here.
2b, and the location of the experimental wake measurement plane is also provided.Since the focus of the present investigation was on the tip gap flow, the hutb endwall was not resolved.Specifically, the k=1 boundary (Figure 2b) was treated as an inviscid symmetry boundary thereby reducing the grid requirements in the spanwise direction.This boundary was situated approximately 2/3 chord inboard of the blade tip.The gap centerline boundary, k=nk, was treated as either an inviscid symmetry boundary or a viscous boundary depending on the whether Experiment A, B or C was simulated.The computations were performed at an inlet Mach number of 0.2 to accommodate the compressible formulation of the flow solver.

A
The flow solver was first applied to Experiment A, detailed above.The clearance gap for which the computations were performed was .04chord (ti/c = .04,see Figure 1).Three calculations were done, including an Euler solution using an embedded H-grid, a Navier-Stokes solution using a "pinched" standard H-grid and a Navier-Stokes solution using the embedded H-grid shown in Figure 2. In Figure 3, cross sectional views of the computational grids used for these three studies are provided.For all three grids, 69 and 35 points were used in the streamwise and spanwise directions, respectively.In the pitchwise direction, 59 grid points were used for the two Navier-Stokes calculations and 45 points were used for the Euler calculation, since very high clustering was not required near the blade surfaces.
As mentioned above, predicted blade pressures and wake flow angle contours from the embedded H-grid and "pinched" standard H-grid Navier-Stokes computations were compared in Basson et al. (1991), and these results are not repeated here.New results of section normal force coefficient and gap mass flow rate are included below for these two Navier-Stokes grid topologies.compared with measured values well away (.33 chord) from the tip.Agreement is good for both numerical solutions, considering that only 35 grid points were used in the streamwise direction on the blade surfaces.At .01 and .08chord inboard of the tip, both computations show the characteristic migration of the suction peak towards midchord.This pressure drop on the suction surface is due to the presence of a leakage vortex adjacent to the suction surface.At both .01 and .08chord inboard of the tip, the Navier-Stokes solution agrees reasonably well with the experimental data.At both locations, the viscous solution underpredicts the magnitude of the suction peak, and predicts the location of this peak approximately 10% chord upstream of the measured peak.Apparently, the Navier-Stokes analysis predicts that the tip vortex is shed somewhat too close to the leading edge.The unloading on the pressure surface is predicted well, especially by the Navier-Stokes code.
The Euler solution also provides qualitatively good predictions of the near tip blade pressure distribution.A suction peak is predicted which is somewhat higher than both the Navier-Stokes and experimental values, and again the location of the suction peak seems to be somewhat too close to the leading edge.Nevertheless, the important feature of increase in suction pressure due to the presence of the tip vortex is captured by the inviscid procedure.
Figure 5a shows blade to blade pressure contours at midspan (k=1) as predicted by the Navier-Stokes code (embedded H-grid).This conventional "two-dimensional" distribution contrasts with pressure contours one grid plane inboard of the tip (k=24), as shown in Figure 5b.Clearly evident is the trajectory of the tip clearance vortex, whose low pressure core is seen to make approximately a 48 0 angle with the axial direction.The core pressure of the decaying vortex is seen to increase approximately linearly with distance along its trajectory up to the cascade exit plane.The core pressure is then seen to increase towards ambient more slowly.Figure 5c provides computed blade to blade pressure contours, one grid plane inboard of the tip (k=24), for the Euler solution.Unlike the blade surface pressure distributions, significant discrepancy between Euler and Navier-Stokes solutions are now seen.Specifically, the tip clearance vortex trajectory is curved in the inviscid simulation.At the cascade exit plane, the low pressure core is seen to make approximately a 600 angle with the axial direction.This turning of the vortex trajectory away from axial may be attributable to increased mass flow through the gap, due in turn to the lack of viscous blockage in the gap.This item is discussed further below.
Measured and computed contours of outlet flow angle are presented in Figure 6.The location of this outlet plane is one-half chord axially downstream of the cascade exit plane, as labelled in Figure 2b.The Navier-Stokes solution using the embedded H-grid is seen to provide good agreement with experiment in prediction of exit flow angle distribution.The tight "binding" of the flow angle contours represents the vortex core.Clearly, the size, strength and trajectory of the tip clearance vortex is captured with good accuracy.The viscous simulation predicts that the vortex is somewhat closer to the gap centerline than the measured vortex.
Outlet flow angle contours predicted by the Euler solution using the embedded H-grid is shown in Figure 6c.The vortex trajectory is seen to be significantly different from the experimental data, as evidenced by the location of the core (compare with Figures 6a and 6b).Also, the range of flow angles is seen to be somewhat higher than the Navier-Stokes solution.This suggests that the Euler code predicts a stronger vortex, consistent with the increase in the magnitude of the suction peak observed in Figures 4a and 4b.As mentioned in the introduction, vorticity generation in subsonic Euler computations relies exclusively on artificial dissipation, whether inherent in the numerical scheme or purposefully added to the -(z + ti) /1   (1990), for example].
In the Euler calculation , the leakage velocities are larger, and hence leakage fluid tends to move somewhat farther away from the suction surface before interacting with the main flow and rolling up into a vortex.Also, as mentioned above, increased leakage velocities tend to convect the vortex towards the pressure surface.These items are further evidenced in Figure 7, which shows contours of normalized dynamic head at the outlet measurement plane, as provided by experimental measurement and Navier-Stokes and Euler solutions.In Figure 7b, the location of the tip vortex at this outlet location is predicted to be close to mid-passage, by the Navier-Stokes solution, in good agreement with experimental observation.This can be seen by comparison of the relative location of low dynamic pressure regions in the vortex core and wake (Note that the computational grid did not exactly follow the wake centerline trajectory, which gave rise to the "drifting" of the wake into the computational domain).The outlet location of the tip vortex, predicted by the Euler solution, is again seen to be in poorer agreement with experiment in Figure 7c.

11
.00 .07.14.21 a Figure 8 shows a plot of the variation of mass weighted, pitchwise averaged flow angle with span, at the outlet measurement plane.Comparison is made in this figure between Navier-Stokes, Euler and measured values.The viscous results are in good agreement with experimental data.Specifically, as the tip gap centerline is approached, average flow angle is seen to decrease from its midspan value (overturning), due to the presence of the vortex.Average flow angle then rapidly increases to a peak of approximately 50 0 , 30 higher than the measured value.The inviscid results show qualitatively good agreement, though again, an overprediction of vortex strength is seen.The Euler procedure shows that the vortex influences the passage flow field as far as 15 %   span inboard of the tip, while data and the Navier-Stokes solution indicate that the influence of the leakage vortex is confined to a region approximately 10 % span inboard of the tip.
For moderate clearances, the decrease in suction side pressure near the tip due to the presence of the vortex can yield a slight increase in section lift inboard of the tip.In Figure 9, the spanwise distribution of normal force coefficient is plotted for Experiment A. Included in this plot are measured values, and computed values using the Euler analysis, the Navier-Stokes analysis using the embedded H-grid and for comparison, the Navier-Stokes analysis using the "pinched" standard H-grid.All three solutions capture qualitatively the increase in section lift, whose peak is located approximately 4 % chord inboard of the tip.The Euler and Navier-Stokes predictions straddle the measured values.This is consistent with previous observations that the Euler and Navier-Stokes solutions slightly overpredict and underpredict, respectively, the strength of the clearance vortex.The section lift provided by the "pinched" standard H-grid Navier-Stokes solution shows some oscillatory behavior near the tip, but is essentially in good agreement with data and the other Navier-Stokes solution.
For the moderate clearance considered, tic = 0.04, not all of the spanwise vorticity bound in the blade is shed at the tip.This gives rise to the so-called "retained lift" at the tip, evident in the measurements in Figure 9.This feature is captured qualitatively in all of the numerical solutions, though the Navier-Stokes solution is seen to provide a less severe drop in CN as the tip is approached.

Experiment B
The previous investigations served to isolate the effect of tip clearance on the passage and downstream flow field without the presence of an endwall or secondary motion due to the transport of an inlet vorticity profile.In an axial compressor, endwall boundary layers on the hub and casing are turned in the blade row.The circumferential force acting on this endwall fluid, of nonuniform streamwise momentum, gives rise to secondary flow.Near the tip, the sense of the secondary motion is opposite that of the tip clearance vortex.This is illustrated schematically in Figure 10, which has been adapted from Lakshminarayana and Horlock (1965).
In Experiment B, the contribution of passage secondary flow was incorporated by generating a normal (to the streamwise direction) vorticity profile upstream of the blade row.This wake (see Figure lb) was generated in the laboratory by inserting a plate along the cascade symmetry plane well upstream of the cascade.In Figure lla, measured total velocity one-half chord upstream of the inlet plane and a short distance downstream of the profile generating plate is shown.This profile was used in constructing inlet boundary conditions for the Navier-Stokes simulation.
In Figures 12a and 12b, average flow angle variation with span is plotted for the cases with no tip gap and with the same gap of 0.04 chord used in Experiment A (distance from gap centerline to tip = 0.04c).For the case with no clearance, the influence of secondary flow is clearly evident.Some underturning is observed as the tip centerline is approached, followed by a more significant average overturning (refer to schematic, Figure 10).These phenomena are very well captured by the Navier-Stokes simulation.The somewhat offsetting effects of endwall secondary flow and tip clearance flow are apparent in Figure 12b.Specifically, the measured and predicted average flow angles are noticeably lower (= 3 0 -40 ) than in Experiment A, where no passage secondary flow was present (see Figure 8).Apparently, the Navier-Stokes simulation underpredicts the influence of the secondary motion somewhat.This is evidenced by an approximately 4 0 overprediction of gap centerline average flow angle, compared to approximately 30 for Experiment A. This may be due to numerical diffusion of the inlet vorticity profile as it is transported through the discretized domain.Comparison of Figures 12a and  12b shows that predicted average midspan flow angle is 1.6° more for Experiment B with no clearance than for the case with clearance.This variation is due to the proximity of the k=1 no-flow-through boundary to the tip (k=1 -2/3 chord inboard of tip).

Figures 13a and 13b
show spanwise distribution of blade normal force coefficient for Experiment B cases with T/c = 0.00 and tic = 0.04.When no clearance is present, a significant increase in section lift is observed near the tip.This is caused by overturning of the flow due to secondary motion.In Figure 13b, where ti/c = 0.04, the normal force distribution is similar to the no clearance case, except near the tip where the blade ends.In both cases, the Navier-Stokes results are remarkably good.The effect of the interaction between secondary flow and leakage flow on near tip loading is captured accurately.These results are encouraging, considering the importance of near tip loading predictions in modem low aspect ratio blading design.

Experiment C
By inserting a thin plate along the gap centerline, Lakshminarayana and Horlock (1965) were able to incorporate the influence of an endwall in their experimental investigation of tip clearance phenomena.In the laboratory, this plate extended from the end of the upstream plate used in Experiment B to the cascade exit plane (see Figure lc).The endwall total velocity profile, measured one-half chord upstream of the cascade, was much steeper than in Experiment B, as shown in Figure llb, due to the presence of the endwall.This experiment served to incorporate the effects of stronger secondary motions and the additional blockage present in the gap itself on the near tip cascade flow field.In the numerical simulation, the measured inflow velocity profile, shown in Figure llb, was used to construct inlet boundary conditions, and the endwall (k=nk, refer to Figure 2b) boundary was changed from a symmetric no-flowthrough boundary to a viscous no-slip boundary.

a
When the cascade was operated with an endwall but with no clearance, a strong secondary flow pattern was observed.In Figures 14a and 14b, measured and computed contours of passage flow angle, a, are shown at the downstream measurement plane.A large region of flow underturning is evident, on the suction side of the wake, which extends well away from the endwall.This is due to a large region of separated flow in the suction surface-endwall corner.This phenomena is common in compressor cascades, stators and rotors, and has been observed by researchers in the absence of leakage flow.In the present cascade, the influence of strong secondary motion itself is apparent in the exit flow angle distribution.Specifically, significant overturning (amin 270) is observed near the endwall at mid-passage in both experiment and Navier-Stokes simulation.

R
Though the effect of secondary motion on flow angle is captured well, the flow angle distribution in the suction surface-endwall separation region is in some error.The location of maximum underturning (-520 ) is predicted to be farther from the endwall and wake centerline.In .Figure 14c, spanwise mass averaged flow angle distributions are plotted for this no clearance case.The general trend of underturning then overturning with span is captured, but the Navier-Stokes solution significantly overpredicts the average overturning near the gap centerline due to the error in predicting the location of the region of underturning arising from flow separation.
In Figures 15a and 15b, the exit flow angles are compared for Experiment C with a tip clearance of 0.04 chord.A number of interesting features are accessible in these figures.The secondary flow is seen to persist in both experiment and Navier-Stokes simulation as evidenced by flow overturning of approximately 5 0 (amin = 27 0 ) near the endwall at midpassage.The presence of the tip clearance vortex is seen to have two effects.The secondary motion "pushes" the leakage vortex towards the suction side (compare with Experiment A, Figure 6).Also, spanwise flow along the suction surface is induced by the tip vortex (see Figure 10), and this injects high momentum fluid into the suction surface-endwall corner, reducing flow separation there.The Navier-Stokes simulation does an excellent job in resolving these complex interactions.This is further evidenced in Figure 15c, which shows that the Navier-Stokes code accurately predicts the spanwise distribution of average flow angle.

Additional Results and Discussion
The above presentation corresponds to comparison of measured and computed flow field parameters and interpretation of these results.In this section, the Navier-Stokes simulations are interrogated further and compared to one another.Additionally, the results of a Navier-Stokes solution, for which experimental measurements could not be taken, are presented.Specifically, Experiment C was rerun with a clearance of 0.04 chord, with simulated endwall rotation.

36.0
The influence of endwall rotation on the clearance flow in turbomachines can be significant.Recently Yaras et al. (1991aYaras et al. ( , 1991b) ) presented their work on the effects of endwall rotation on the clearance flow in turbines.They performed experimental investigations where endwall rotation was simulated in a turbine cascade by rotating an endwall "belt" adjacent to the blade tips.They observed many interesting phenomena due to the endwall rotation, including reduced mass flow rate through the gap, and transport of passage and tip clearance vortices back toward the suction side of the wake.(1991a, 1991b) experimental turbine cascade studies.The r r r "--. .t present simulation of wall motion corresponds to that of a .4-..t 'N 'N compressor rotor with a practical value of flow `^\ `^coefficient.Specifically, along the entire endwall .
`^boundary, k = nk (see Figure 2b), a pitchwise velocity of I C 0.5 qin-midspan was imposed.Aside from this, the code was run as for Experiment C, with ti/c = 0.04.The ` ^``` ``results of this study are included in the presentations below.
In Figure 16, comparisons of spanwise average flow angle distribution are made between the various Navier-Stokes solutions.Figure 16a shows the influence of secondary flow strength on flow angle in cases with moderate tip clearance.The influence of weak secondary flow (Experiment B) is seen to decrease net underturning due to the leakage flow.When strong secondary flow is present (Experiment C) the leakage vortex induced underturning is reduced significantly by the overturning due to secondary motion.Alternatively, the influence of flow separation on the suction surface provides more underturning well away from the tip gap [ -(z + t) / 1 0.04 to 0.10 ] than in Experiments A and B, where suction side-endwall separation was zero and small respectively.
The influence of this flow separation is very evident in the cases with no clearance, as shown in Figure 16b.In the case of weak secondary flow, average flow angle increases slightly with span due to a small suction surface-endwall separation zone discussed earlier.In Experiment C, the effect of flow separation is more significant as seen by increased undertuming away from the gap centerline.In Figure 16c, the three Experiment C results, with different clearances and wall motion, are compared.Clearly captured are the average underturning near the gap centerline due to the tip vortex and the underturning due to flow separation away from the gap centerline.In the case with simulated endwall rotation, the endwall boundary layer is seen to skew significantly and the average flow angle approaches 90 ° at the endwall.
The relative motion between the blade and the endwall augments the leakage flow in the case of a compressor, giving rise to higher underturning angles near the wall [ -(z + t) / 1 = 0.00 to 0.04 1.The leakage vortex is convected by the wall motion towards the pressure surface.This along with the reduced endwall flow separation decreases underturning in the region [ -(z + t) / 1 > 0.04 ] from those observed without wall motion.
In Figure 17, a plot of flow velocity vectors in the gap region at midchord for several of the computed cases is provided.These figures are included to provide some qualitative indication of the effects of mesh type and clustering, viscosity and endwall treatment on the nature of the computed flow field in the gap.The embedded Hgrid Navier-Stokes solutions are seen to provide qualitative development of boundary layers on the tip surface and endwall when present.These boundary layers reduce the leakage mass flow and strength of the tip vortex.The leakage velocities are generally higher for the Euler case and, of course, no tip surface or endwall boundary layers develop.All computations correctly predict the outward flow near the pressure surface towards the tip.It is noted that the imposed endwall velocity ( = 0.5 qin-midspan) is approximately 1/2 the peak gap cross flow velocity for the case with endwall motion.
In Figures 18, the mass flow rate through the gap is plotted for the three Experiment A computations.In Figure 18a, accumulated mass flow is plotted against chord for Euler, Navier-Stokes with embedded H-grid and Navier-Stokes with "pinched" standard H-grid calculations.Clearly the mass flow rate predicted by the Euler computation is significantly higher than that of the Navier-Stokes solutions.The inviscid solution predicts a total mass flow rate through the gap of approximately 2.3 % of inlet mass flow rate compared to 1.6 % for the embedded H-grid Navier-Stokes solution and 1.8 % for the "pinched" grid Navier-Stokes solution.These observations are consistent with the flow vectors plotted in Figure 17, and the previous observation that the angle that the Euler vortex trajectory makes with axial is too large.In Figure 18b, the normalized local mass flow rate through the gap is plotted against chord.All three solutions agree qualitatively, but the peak mass flow rate predicted by the Euler solution is closer to 2/3 chord, while the embedded H-grid Navier-Stokes procedure provides a more realistic location of approximately 1/2 chord.
Figure 19 compares embedded H-grid Navier-Stokes mass flow rate predictions for the four experiments run with clearance (A, B, C, C with simulated endwall rotation).In Figure 19a, total mass flow through the gap is seen to decrease slightly with a weak secondary flow present (compare Experiments A and B), and is seen to decrease dramatically when an endwall is present with stronger secondary motion (Experiment C).The decrease in mass flow due to an endwall arises from increased blockage in the gap (see flow vectors, Figure 17).Decrease in mass flow is also due to blockage induced by secondary motion, which is oppositely directed to the gap flow near the endwall.Simulated endwall rotation increases mass flow through the gap for Experiment C, which is as expected.In       In Figure 20,, contours of dynamic pressure at the outlet measurement plane are provided for Experiments C. For the two cases with no endwall motion, comparison is made with experimental values.With no clearance (Figure 20a), a high loss region is seen near the gap centerline close to the suction side of the cascade wake.This corresponds to the aforementioned separation zone which develops at the suction surface-endwall interface near the trailing edge of the blade row.Consistent with the flow angle distribution for this case (refer to Figure 14), the predicted location of minimum dynamic head is located slightly more inboard of the tip than the measured location.The magnitude of losses in dynamic head due to secondary flow and flow separation are predicted reasonably well.For the case with clearance (Figure 20b), a distinct vortex is evidenced.This vortex is seen to reduce the extent of the high loss, low dynamic head region since spanwise flow of high momentum fluid is induced along the suction surface (see Figure 10).In this case, the prediction of interactions between and mixing losses due to leakage flow, secondary flow and separation are predicted well, including the location of the vortex core.
The computed contours of normalized dynamic pressure for the case with simulated endwall rotation is shown in Figure 20c.Clearly the motion of the endwall has caused the tip clearance vortex to be carried towards midpassage.Analogous to Yaras ' et al. (1991b) results and discussions, modification of the vortex trajectory is due in part to the increased (decreased in Yaras' turbine cascade) mass flow rate through the gap (inviscid effect) and in part to the viscous dragging effect induced by the endwall motion.The vortex is seen to be stretched due to endwall motion.
In Figure 21, contours of local vorticity dotted with the midspan mean flow velocity () ' q2D) are presented at four chordwise locations for Experiment C, with tic = 0.04 and no endwall motion.These plots provide qualitative indication of the nature of the developing tip clearance vortex and its interaction with the passage sc.^ondary motion and vortex.At 1/3 chord, the tip vortex has begun to form, its pitchwise width approximately equal to the local blade thickness.Also evident is a small and comparatively weak counterrotating vortex (secondary vortex) inboard of and further away from the suction surface than the forming tip vortex.By 2/3 chord, an identifiable core has formed as the clearance vortex has grown and moved both inboard of the tip and away from the suction surface.The other vortex has diffused but remains adjacent to the tip vortex.Endwall secondary flow is seen to interact with the leakage vortex at the cascade exit plane as evidenced by positive values of w ' q2D near the endwall and pressure surfaces.The leakage vortex has grown in diameter to approximately 2/5 cascade pitch at this trailing edge plane.At the outlet measurement plane, the tip vortex has diffused somewhat, but, as discussed above, its pitchwise location is seen to have been restricted to the suction side of the wake by the secondary motion.

CONCLUSION
A series of computational simulations of tip clearance flow in a compressor cascade has been undertaken using a Navier-Stokes/Euler code that can accommodate a grid topology which explicitly resolves the flow in the tip gap region.The following observations and conclusions apply: 1) The inviscid analysis provided qualitative representation of some of the important physical phenomena in a compressor cascade flow with tip clearance, including blade unloading due to the formation of a tip clearance vortex.The Navier-Stokes simulation was seen to provide better agreement with the experimental data in predicting vortex trajectory and strength, and is presumed to provide more accurate tip gap mass flow rate predictions.
2) Additional results were presented which suggest that a "pinched" standard H-grid topology can provide good engineering approximation in viscous flow computations with tip clearance.However, significant discrepancy in predicted tip gap mass flow rate was observed between solutions which incorporated the two topologies.
3) The embedded H-grid Navier-Stokes procedure was shown to provide good representation of the dominant physical mechanisms in turbomachinery flows with tip clearance, including the strength and trajectory of the tip clearance vortex, blade loading phenomena including lift retained at the tip, and the interaction of passage secondary flow and tip clearance flow.This was evidenced by comparison of the results of several parametric investigations with available experimental measurements of flow angle, blade loading and dynamic head in the cascade investigated.
4) The several Navier-Stokes solutions proved useful in interpreting the nature of the complex interactions in the tip clearance region, provided simulation of the effect of endwall rotation and yielded information such as mass flow rate through the gap, not easily accessible experimentally.
In the opinion of the authors, the use of a Navier-Stokes analysis, such as that used herein, can provide engineering accuracy in the determination of many design critical flow field features and parameters in turbomachinery stages with tip clearance.These include blade loading, discharge flow rates and the nature of the outlet flow (flow angles, losses) seen by subsequent stages.

Figure 1 .
Figure 1.Schematic of experimental configurations.(Note: z = 0 is located .04 chord inboard of the cascade symmetry plane [k=nk].For the cases with clearance, this corresponds to the blade tip [tic = .04])

Figure 2 .
Figure 2. Views of computational mesh and nomenclature a) Detail of leading edge.b) View of three-dimensional grid.

Figures 4
Figures 4 show predicted Navier-Stokes (embedded H-grid) and Euler blade surface pressure distribution at spanwise locations very near the tip (z/c = -.01),near the tip (z/c = -.08) and far from the tip (z/c = -.33).In Figure 4c, Euler and Navier-Stokes solutions are

Figure 4 .
Figure 4. Comparison of computed and measured blade static pressure coefficient at three spanwise locations.Measured values (symbols), Navier-Stokes (solid line), Euler (short dash line).

Figure 5 .
Figure 5. Blade to blade contours of static pressure(kPa) predicted by the Navier-Stokes and Euler simulations.a) Navier-Stokes, midspan grid slice.b) Navier-Stokes, grid slice just inboard of tip.c) Euler, grid slice just inboard of tip.

Figure 6 .
Figure 6.Contours of flow angle, a (degrees) at outlet

Figure 15 .
Figure 15.Contours of flow angle at outlet measurement plane for Experiment C, r/c = 0.04.a) Measured.b) Predicted.c) Comparison of computed and measured spanwise distribution of mass-weighted, pitchwise averaged outlet flow angle for Experiment C, ti/c = 0.04.Measured values (symbols), Navier-Stokes

Figure 18 .
Figure 18.Comparison of predicted mass flow rates through the gap for Experiment A using Euler, Navier-Stokes with an embedded H-grid and Navier-Stokes with a "pinched" standard H-grid.a) Accumulated mass flow through the gap.b) Variation of incremental mass flow rate through the gap with chord.

Figure 19 .
Figure 19.Comparison of predicted mass flow rates through the gap for Experiment A with tic = 0.04, Experiment B with tic = 0.04, Experiment C with tic = 0.04 and Experiment C with tic = 0.04 and simulated endwall rotation.a) Accumulated mass flow through the gap.b) Variation of incremental mass flow rate through the gap with chord.

Figure 19b ,
Figure19b, normalized local mass flow rate through the gap is plotted against chord for these four Navier-Stokes solutions.